Neural-network-based multistate solver for a static Schrödinger equation

Solving a multivariable static Schr\"odinger equation for a quantum system, to produce multiple excited-state energy eigenvalues and wave functions, is one of the basic tasks in mathematical and computational physics. Here we propose a neural-network-based solver, which enables us to cover the high-dimensional variable space for this purpose. The efficiency of the solver is analyzed by examples aimed at demonstrating the concept and various aspects of the task: the simultaneous finding of multiple excited states of lowest energies, the computation of energy-degenerate states with orthogonalized wave functions, the scalability to handle a multivariable problem, and the self-consistent determination and automatic adjustment of the imbedded Monte Carlo procedure. The solver adheres to the computational techniques developed in machine learning and is vastly different from traditional numerical methods.